Optimal. Leaf size=849 \[ 12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )-6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_4\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_4\left (\sqrt {-d} \sqrt {f} x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.70, antiderivative size = 849, normalized size of antiderivative = 1.00, number
of steps used = 26, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules
used = {2342, 2341, 2425, 209, 2361, 12, 4940, 2438, 2367, 2354, 2421, 6724, 2430}
\begin {gather*} 12 b^3 \sqrt {d} \sqrt {f} \text {ArcTan}\left (\sqrt {d} \sqrt {f} x\right ) n^3-\frac {6 b^3 \log \left (d f x^2+1\right ) n^3}{x}-6 i b^3 \sqrt {d} \sqrt {f} \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right ) n^3+6 i b^3 \sqrt {d} \sqrt {f} \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right ) n^3+6 b^3 \sqrt {-d} \sqrt {f} \text {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) n^3-6 b^3 \sqrt {-d} \sqrt {f} \text {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) n^3-6 b^3 \sqrt {-d} \sqrt {f} \text {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right ) n^3+6 b^3 \sqrt {-d} \sqrt {f} \text {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right ) n^3+12 b^2 \sqrt {d} \sqrt {f} \text {ArcTan}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right ) n^2-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d f x^2+1\right ) n^2}{x}-6 b^2 \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) n^2+6 b^2 \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) n^2+6 b^2 \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) n^2-6 b^2 \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) n^2+3 b \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right ) n-3 b \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt {-d} \sqrt {f} x+1\right ) n-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) n}{x}-3 b \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) n+3 b \sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) n+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt {-d} \sqrt {f} x+1\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 2341
Rule 2342
Rule 2354
Rule 2361
Rule 2367
Rule 2421
Rule 2425
Rule 2430
Rule 2438
Rule 4940
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx &=-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-(2 f) \int \left (-\frac {6 b^3 d n^3}{1+d f x^2}-\frac {6 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}{1+d f x^2}-\frac {3 b d n \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2}\right ) \, dx\\ &=-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}+(2 d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2} \, dx+(6 b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx+\left (12 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{1+d f x^2} \, dx+\left (12 b^3 d f n^3\right ) \int \frac {1}{1+d f x^2} \, dx\\ &=12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}+(2 d f) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 \left (1-\sqrt {-d} \sqrt {f} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 \left (1+\sqrt {-d} \sqrt {f} x\right )}\right ) \, dx+(6 b d f n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1-\sqrt {-d} \sqrt {f} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1+\sqrt {-d} \sqrt {f} x\right )}\right ) \, dx-\left (12 b^3 d f n^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f} x} \, dx\\ &=12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}+(d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1-\sqrt {-d} \sqrt {f} x} \, dx+(d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1+\sqrt {-d} \sqrt {f} x} \, dx+(3 b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1-\sqrt {-d} \sqrt {f} x} \, dx+(3 b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt {-d} \sqrt {f} x} \, dx-\left (12 b^3 \sqrt {d} \sqrt {f} n^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx\\ &=12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-\left (3 b \sqrt {-d} \sqrt {f} n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx+\left (3 b \sqrt {-d} \sqrt {f} n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{x} \, dx-\left (6 b^2 \sqrt {-d} \sqrt {f} n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx+\left (6 b^2 \sqrt {-d} \sqrt {f} n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{x} \, dx-\left (6 i b^3 \sqrt {d} \sqrt {f} n^3\right ) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx+\left (6 i b^3 \sqrt {d} \sqrt {f} n^3\right ) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx\\ &=12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )-6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\left (6 b^2 \sqrt {-d} \sqrt {f} n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx-\left (6 b^2 \sqrt {-d} \sqrt {f} n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx+\left (6 b^3 \sqrt {-d} \sqrt {f} n^3\right ) \int \frac {\text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx-\left (6 b^3 \sqrt {-d} \sqrt {f} n^3\right ) \int \frac {\text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx\\ &=12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )-6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-\left (6 b^3 \sqrt {-d} \sqrt {f} n^3\right ) \int \frac {\text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx+\left (6 b^3 \sqrt {-d} \sqrt {f} n^3\right ) \int \frac {\text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx\\ &=12 b^3 \sqrt {d} \sqrt {f} n^3 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )-6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+6 i b^3 \sqrt {d} \sqrt {f} n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_4\left (-\sqrt {-d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \text {Li}_4\left (\sqrt {-d} \sqrt {f} x\right )\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 794, normalized size = 0.94 \begin {gather*} 2 \sqrt {d} \sqrt {f} \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-\frac {\left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 b \left (a^2+2 a b n+2 b^2 n^2\right ) \log \left (c x^n\right )+3 b^2 (a+b n) \log ^2\left (c x^n\right )+b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}+3 i b \sqrt {d} \sqrt {f} n \left (a^2+2 a b n+2 b^2 n^2+2 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )-\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )+6 i b^2 \sqrt {d} \sqrt {f} n^2 \left (a+b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\frac {1}{2} \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{2} \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-\text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )+i b^3 \sqrt {d} \sqrt {f} n^3 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )-6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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